Automorphisms of certain Stein manifolds (Q1897993)

Let \((M,g)\) be a compact, real analytic Riemannian manifold. Let \(0<r\leq\infty\). Denote by \(T^rM\) the disk bundle over \(M\) that consists of all the tangent vectors having \(g\)-length smaller than \(r\). For a given \((M,g)\) there exists an \(r\) and a canonically defined (adapted to the metric \(g\)) complex manifold structure \(J_g\) on \(T^rM\). The \(g\)-norm square function \(\rho\) is strictly plurisubharmonic with respect to \(J_g\), hence \((T^rM,J_g)\) is a Stein manifold. Denote by \(\kappa_g\) the Kähler metric on \(T^rM\) induced by \(\rho\). The main purpose of this paper is to study biholomorphic maps between and automorphisms of this type of Stein manifolds. We also discuss the question when such a biholomorphism is an isometry with respect to \(\kappa_g\).